Séminaire Lotharingien de Combinatoire 52 (2004), Article B52h

SOME CONJECTURES FOR

MACDONALD POLYNOMIALS OF TYPE B, C, D

Michel Lassalle

Centre National de la Recherche Scientifique Institut Gaspard Monge, Université de Marne-la-Vallée 77454 Marne-la-Vallée Cedex, France

Dedicated to Alain Lascoux on the occasion of his 60th birthday

Abstract. We present conjectures giving formulas for the Macdonald polynomials of type B, C, D which are indexed by a multiple of the first fundamental weight. The transition matrices between two different types are explicitly given.

Introduction

Among symmetric functions, the special importance of Schur functions comes from their intimate connection with representation theory. Actually the irreducible polynomial representations of GL_{n}(ℂ) are indexed by partitions λ = (λ_{1}, …, λ_{n}) of length ≤ n, and their characters are the Schur functions s_{λ}.

In the eighties, I. G. Macdonald introduced a new family of symmetric functions P_{λ}(q, t). These orthogonal polynomials depend rationally on two parameters q, t and generalize Schur functions, which are obtained for t = q [9,10].

When the indexing partition is reduced to a row (k) (i.e. has length one), the Macdonald polynomial g_{k}(q, t) of n variables x = (x_{1}, …, x_{n}) are given by their generating function

∏_{i=1}^{n} \frac{(tux_{i}; q)_{∞}}{(ux_{i}; q)_{∞}} = ∑_{r≥0} u^{r}g_{r}(x; q, t),

with the standard notation (a; q)_{∞} = ∏_{i=0}^{∞}(1 − aq^{i}). Of course for t = q the complete functions s_{(r)} = h_{r} are recovered.

A few years later, generalizing his previous work, Macdonald introduced another class of orthogonal polynomials, which are Laurent polynomials in several variables, and generalize the Weyl characters of compact simple Lie groups [11,12]. In the most simple situation of this new framework, a family P_{λ}^{(R)}(q, t) of polynomials, depending rationally on two parameters q, t, is attached to each root system R.

Typeset by AMS-TEX

These orthogonal polynomials are elements of the group algebra of the weight lattice of R, invariant under the action of the Weyl group. They are indexed by the dominant weights of R.

When R is of type A, the orthogonal polynomials P_{λ}^{(R)}(q, t) correspond to the symmetric functions P_{λ}(q, t) previously studied in [9,10]. For t = q, they correspond to the Weyl characters χ_{λ}^{(R)} of compact simple Lie groups.

This paper is only devoted to the Macdonald polynomials which are indexed by a multiple of the first fundamental weight ω_{1}. Since H. Weyl [15], it is well known that χ_{rω_{1}}^{(R)} is given by

(i) h_{r}(X) + h_{r−1}(X), when R = B_{n},

(ii) h_{r}(X), when R = C_{n},

(iii) h_{r}(X) − h_{r−2}(X), when R = D_{n},

with X = (x_{1}, …, x_{n}, 1/x_{1}, …, 1/x_{n}).

However, as far as the author is aware, no such result is known when t ≠ q, and no explicit expansion is available for the Macdonald polynomials P_{rω_{1}}^{(R)}(q, t). The purpose of this paper is to present some conjectures generalizing the previous formulas.

Actually this problem can be considered in a more general setting, allowing two distinct parameters t, T, each of which is attached to a length of roots. We give an explicit formula for P_{rω_{1}}^{(R)}(q, t, T) when R is of type B, C, D, together with an explicit formula for the transition matrices between different types. The entries of these transition matrices appear to be fully factorized and reveal deep connections with basic hypergeometric series.

We provide some support for these conjectures by showing that they are verified upon principal specialization. On the other hand, computer calculations show a very strong empirical evidence in their favor.

1. Macdonald polynomials

In this section we introduce our notations, and recall some general facts about Macdonald polynomials. For more details the reader is referred to [11,12,13].

The most general class of Macdonald polynomials is associated with a pair of root systems (R, S), spanning the same vector space and having the same Weyl group, with R reduced. Here we shall only consider the case of a pair (R, R), with R of type B, C, D.

Let V be a finite-dimensional real vector space endowed with a positive definite symmetric bilinear form ⟨u, v⟩. For all v ∈ V, we write |v| = ⟨v, v⟩^{1/2} , and v^{∨} = 2v/|v|^{2}.

Let R ⊂ V be a reduced irreducible root system, W the Weyl group of R, R^{+} the set of positive roots, {α_{1}, …, α_{n}} the basis of simple roots, and R^{∨} = {α^{∨} | α ∈ R} the dual root system.

Let Q = ∑_{i=1}^{n} ℤ α_{i} and Q^{+} = ∑_{i=1}^{n} ℕ α_{i} be the root lattice of R and its positive octant. Let P = {λ ∈ V | ⟨λ, α^{∨}⟩ ∈ ℤ ∀α ∈ R} and P^{+} = {λ ∈ V | ⟨λ, α^{∨}⟩ ∈ ℕ ∀α ∈ R^{+}} be the weight lattice of R and the cone of dominant weights.

A basis of Q is formed by the simple roots α_{i}. A basis of P is formed by the fundamental weights ω_{i} defined by ⟨ω_{i}, α_{j}^{∨}⟩ = δ_{ij}. A partial order is defined on P by λ ≥ μ if and only λ − μ ∈ Q^{+}.

Let A denote the group algebra over ℝ of the free Abelian group P. For each λ ∈ P let e^{λ} denote the corresponding element of A, subject to the multiplication rule e^{λ}e^{μ} = e^{λ+μ}. The set {e^{λ}, λ ∈ P } forms an ℝ-basis of A.

The Weyl group W acts on P and on A. Let A^{W} denote the subspace of W - invariants in A. Such elements are called “symmetric polynomials”. There are two important examples of a basis of A^{W}. The first one is given by the orbit-sums

m_{λ} = ∑_{μ∈W λ} e^{μ}, λ ∈ P^{+}.

Another basis is provided by the Weyl characters defined as follows. Let

δ = ∏_{α∈R^{+}}(e^{α/2} − e^{−α/2}) = e^{−ρ} ∏_{α∈R^{+}}(e^{α} − 1),

with ρ = \frac{1}{2} ∑_{α∈R^{+}} α ∈ P. Then wδ = ε(w)δ for any w ∈ W, where ε(w) = det(w) = ±1. For all λ ∈ P,

χ_{λ} = δ^{−1} ∑_{w∈W} ε(w)e^{w(λ+ρ)}

is in A^{W}, and the set {χ_{λ}, λ ∈ P^{+}} forms an ℝ-basis of A^{W}.

Let 0 < q < 1. For any indeterminate x and for all k ∈ ℕ, define

(x; q)_{∞} = ∏_{i=0}^{∞}(1 − xq^{i}), (x; q)_{k} = ∏_{i=0}^{k−1}(1 − xq^{i}).

For each α ∈ R let t_{α} = q^{k_{α}} be a positive real number such that t_{α} = t_{β} if |α| = |β|. Then we have at most two different values for the t_{α}’s. Define

ρ_{k} = \frac{1}{2} ∑_{α∈R^{+}} k_{α}α, ρ_{k}^{∗} = \frac{1}{2} ∑_{α∈R^{+}} k_{α}α^{∨}.

If f = ∑_{λ∈P} a_{λ}e^{λ} ∈ A, let \overline{f} = ∑_{λ∈P} a_{λ}e^{−λ} and [ f ]_{1} its constant term a_{0}. The inner product defined on A by

⟨f, g⟩_{q,t} = \frac{1}{|W |}[fḡΔ_{q,t}]_{1},

with |W | the order of W, and

Δ_{q,t} = ∏_{α∈R} \frac{(e^{α}; q)_{∞}}{(t_{α}e^{α}; q)_{∞}}

is non degenerate and W -invariant.

There exists a unique basis {P_{λ}, λ ∈ P^{+}} of A^{W}, called Macdonald polynomials, such that

(i) P_{λ} = m_{λ} + ∑_{μ∈P^{+}, μ<λ} a_{λμ}(q, t) m_{μ} where the coefficients a_{λμ}(q, t) are rational functions of q and the t_{α}’s,

(ii) ⟨P_{λ}, P_{μ}⟩_{q,t} = 0 if λ ≠ μ.

It is clear that the P_{λ}, if they exist, are unique. Their existence is proved as eigenvectors of an operator E : A^{W} → A^{W}, selfadjoint with respect to ⟨ , ⟩_{q,t}, and having its eigenvalues all distinct. When R is of type B, C, D this operator may be constructed as follows [11,12].

Let π be a minuscule weight of R^{∨}, i.e. a vector π ∈ V such that ⟨π, α⟩ takes only values 0 and 1 for α ∈ R^{+}. Such a vector exists when R is of type B, C, D, and is necessarily a fundamental weight of R^{∨}. Let

Φ_{π} = ∏_{α∈R^{+}} \frac{1 − t_{α}^{⟨π,α⟩}e^{α}}{1 − e^{α}} ,

and T_{π} the translation operator defined on A by T_{π}(e^{λ}) = q^{⟨λ,π⟩}e^{λ} for any λ ∈ P. Let E_{π} the operator defined by

E_{π}f = ∑_{w∈W} w(Φ_{π} · T_{π}f).

Macdonald polynomials P_{λ} are then introduced as eigenvectors of E_{π} :

E_{π} P_{λ} = c_{λ} P_{λ}  with  c_{λ} = q^{⟨π,ρ_{k}⟩} ∑_{w∈W} q^{⟨wπ,λ+ρ_{k}⟩}.

We may regard any f = ∑_{λ∈P} f_{λ}e^{λ} ∈ A, having only finitely many nonzero coefficients, as a function on V by putting for any x ∈ V,

f(x) = ∑_{λ∈P} f_{λ}q^{⟨λ,x⟩}.

Then Macdonald polynomials satisfy the following Specialization Formula [2]

P_{λ}(ρ_{k}^{∗}) = q^{−⟨λ,ρ_{k}^{∗}⟩} ∏_{α∈R^{+}} \frac{(q^{⟨ρ_{k},α^{∨}⟩}t_{α}; q)_{⟨λ,α^{∨}⟩}}{(q^{⟨ρ_{k},α^{∨}⟩}; q)_{⟨λ,α^{∨}⟩}} .

In the sequel we shall consider R to be of type B_{n}, C_{n} or D_{n}. We shall identify V with ℝ^{n} and write ε_{1}, …, ε_{n} for its standard basis. Defining x_{i} = e^{ε_{i}} for i = 1 … n, we shall regard P_{λ} as a Laurent polynomial of n variables x_{1}, …, x_{n}.

We shall only consider Macdonald polynomials P_{λ} associated with a weight λ = rω_{1}, multiple of the first fundamental weight ω_{1} = ε_{1}.

2. Basic hypergeometric series

We shall need three results about basic hypergeometric series. The author is deeply indebted to Professor Mizan Rahman for communicating their proofs to him. Since these results have intrinsic interest, their proofs are given below in Section 10.

We adopt the notation of [3] and write

r+1ϕ_{r} [ \begin{matrix}a_{1}, a_{2}, …, a_{r+1} \\ b_{1}, b_{2}, …, b_{r} \end{matrix}; q, z] = ∑_{i≥0} \frac{(a_{1}; q)_{i} … (a_{r+1}; q)_{i}}{(b_{1}; q)_{i} … (b_{r}; q)_{i}} \frac{z^{i}}{(q; q)_{i}} .

Theorem 1. We have the following transformation between _{2}ϕ_{1} series

\frac{(u; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, ux \\ q^{1−r}/ux \end{matrix}; q, qv/u^{2}x] = ∑_{i=0}^{[r/2]} \frac{(u; q)_{r−2i}}{(q; q)_{r−2i}} _{2}ϕ_{1} [ \begin{matrix}q^{2i−r}, uy \\ q^{1+2i−r}/uy \end{matrix}; q, qv/u^{2}y]

× y^{i}v^{i} \frac{(x/y; q)_{i}}{(q; q)_{i}} \frac{(uq^{r−2i}; q)_{2i}}{(uxq^{r−i}; q)_{i} (uyq^{r−2i+1}; q)_{i}} .

An infinite-dimensional matrix (f_{ij})_{i,j∈ℤ} is said to be lower-triangular if f_{ij} = 0 unless i ≥ j. Two infinite-dimensional lower-triangular matrixes (f_{ij})_{i,j∈ℤ} and (g_{kl})_{k,l∈ℤ} are said to be mutually inverse if ∑_{i≥j≥k} f_{ij}g_{jk} = δ_{ik}.

Corollary. Defining

M_{r,r−2i}^{(u,v;x,y)} = y^{i}v^{i} \frac{(x/y; q)_{i}}{(q; q)_{i}} \frac{(uq^{r−2i}; q)_{2i}}{(uxq^{r−i}; q)_{i} (uyq^{r−2i+1}; q)_{i}} ,

the infinite matrices M(u, v; x, y) and M(u, v; y, x) are mutually inverse.

Michael Schlosser remarked that this corollary is equivalent with Bressoud’s matrix inverse [1], which states that, defining

A_{ij}^{(u,v)} = (u/v)^{j} \frac{(u/v; q)_{i−j}}{(q; q)_{i−j}} \frac{(u; q)_{i+j}}{(vq; q)_{i+j}} \frac{1 − vq^{2j}}{1 − v} ,

the matrices A(u, v) and A(v, u) are mutually inverse. Indeed let d be either 0 or 1. Replacing r by 2r + d, and i by r − i, some factors cancel or can be pulled out of the previous sum, yielding the above form of [1].

Bressoud’s matrix inverse was originally derived from the terminating very-well-poised _{6}ϕ_{5} summation [3, (II.21)]. We refer to [4, 8] for some generalizations of [1], as well as more details and references about inversion of infinite matrices.

Theorem 2. Defining

\frac{(u; q)_{r}}{(q; q)_{r}} H_{r} = ∑_{i=0}^{r} a^{r−i} \frac{(u; q)_{r−i}}{(q; q)_{r−i}} _{2}ϕ_{1} [ \begin{matrix}q^{i−r}, v \\ q^{1+i−r}/v \end{matrix}; q, q/v^{2}a^{2}]

× \frac{(x; q)_{i}}{(q; q)_{i}} \frac{(uq^{r−i}; q)_{i}}{(vq^{r−i+1}; q)_{i}} \frac{(v^{2}q^{2r−i+1}; q)_{i}}{(xv^{2}q^{2r−i}; q)_{i}} ,

we have

H_{r} = (−1/v)^{r} \frac{(xv^{2}; q)_{r}}{(xv^{2}; q)_{2r}} \frac{(x^{2}v^{2}; q^{2})_{r}(v^{2}q; q^{2})_{r}}{(xv; q)_{r}} _{4}ϕ_{3} [ \begin{matrix}q^{−r}, xv^{2}q^{r}, −av, −1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −xv \end{matrix}; q, q] .

A converse property may be stated as follows.

Theorem 3. We have

a^{r} \frac{(u; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, v \\ q^{1−r}/v \end{matrix}; q, q/v^{2}a^{2}] = ∑_{i=0}^{r} \frac{(u; q)_{r−i}}{(q; q)_{r−i}} H_{r−i}

× x^{i} \frac{(1/x; q)_{i}}{(q; q)_{i}} \frac{(uq^{r−i}; q)_{i}}{(vq^{r−i+1}; q)_{i}} \frac{(v^{2}q^{2r−2i+1}; q)_{i}}{(xv^{2}q^{2r−2i+1}; q)_{i}} \frac{1 − v^{2}q^{2r}}{1 − v^{2}q^{2r−i}} .

Observe that, as a consequence of Theorems 2 and 3, we recover the special case y = 1 of the following matrix inverse.

Lemma 1. Defining

N_{ij}^{(u,v;x,y)} = y^{i−j} \frac{(x/y; q)_{i−j}}{(q; q)_{i−j}} \frac{(uq^{j}; q)_{i−j}}{(vq^{j+1}; q)_{i−j}} \frac{(v^{2}q^{2j+1}; q)_{2i−2j}}{(xv^{2}q^{i+j}; q)_{i−j} (yv^{2}q^{2j+1}; q)_{i−j}} ,

the infinite matrices N (u, v; x, y) and N (u, v; y, x) are mutually inverse.

Proof. Defining

B_{ij}^{(x,y)} = y^{i−j} \frac{(x/y; q)_{i−j}}{(q; q)_{i−j}} \frac{1}{(xq^{i+j}; q)_{i−j} (yq^{2j+1}; q)_{i−j}} ,

the infinite matrices B(x, y) and B(y, x) are mutually inverse, as a consequence of a result of Krattenthaler [4]. If two infinite matrices (f_{ij}) and (g_{kl}) are mutually inverse, for any sequence (d_{k}), the matrices (f_{ij} d_{i}/d_{j}) and (g_{kl} d_{k}/d_{l}) are obviously mutually inverse. Since

(v^{2}q^{2j+1}; q)_{2i−2j} \frac{(uq^{j}; q)_{i−j}}{(vq^{j+1}; q)_{i−j}} = \frac{(v^{2}q; q)_{2i}}{(v^{2}q; q)_{2j}} \frac{(u; q)_{i}}{(u; q)_{j}} \frac{(vq; q)_{j}}{(vq; q)_{i}} ,

we apply this property to B(xv^{2}, yv^{2}) and B(yv^{2}, xv^{2}), with

d_{k} = v^{2k} (v^{2}q; q)_{2k} \frac{(u; q)_{k}}{(vq; q)_{k}} . □

3. Type C

For the type C root system, the set of positive roots is the union of R_{1} = {ε_{i} ± ε_{j}, 1 ≤ i < j ≤ n} and R_{2} = {2ε_{i}, 1 ≤ i ≤ n}. Elements of each set have the same length, and we write t_{α} = t for α ∈ R_{1} and t_{α} = T for α ∈ R_{2}.

The Weyl group W is the semi-direct product of the permutation group S_{n} by (ℤ/2ℤ)^{n}. It acts on V by signed permutation of components. The fundamental weights are given by ω_{i} = ∑_{j=1}^{i} ε_{j}, 1 ≤ i ≤ n. The dominant weights λ ∈ P^{+} can be identified with vectors λ = ∑_{i=1}^{n} λ_{i}ε_{i}, such that (λ_{1}, λ_{2}, …, λ_{n}) is a partition. There is only one minuscule weight ω_{1}. The partial order λ ≥ μ is given by

∑_{i=1}^{j}(λ_{i} − μ_{i}) ∈ ℕ, for j = 1, …, n − 1,  ∑_{i=1}^{n}(λ_{i} − μ_{i}) ∈ 2ℕ.

With t = q^{k} and T = q^{K}, we have

ρ_{k} = ∑_{i=1}^{n} ((n − i)k + K) ε_{i}, ρ_{k}^{∗} = ∑_{i=1}^{n} ((n − i)k + K/2) ε_{i}.

If we write x_{i} = e^{ε_{i}} for i = 1 … n, and define

P_{rω_{1}}^{(C)}(x; q, t, T) = \frac{(q; q)_{r}}{(t; q)_{r}} g_{r}^{(C)}(x; q, t, T),

the Specialization Formula reads

g_{r}^{(C)}(t^{n−1}T^{\frac{1}{2}} , t^{n−2}T^{\frac{1}{2}} , …, T^{\frac{1}{2}} ; q, t, T) = \frac{t^{r(1−n)}}{T^{r/2}} \frac{(t^{n}; q)_{r}}{(q; q)_{r}} \frac{(T^{2}t^{2n−2}; q)_{r}}{(T t^{n−1}; q)_{r}} .

Let us introduce the auxiliary quantities G_{r}(x; q, t) defined by their generating function

∏_{i=1}^{n} \frac{(tux_{i}; q)_{∞}}{(ux_{i}; q)_{∞}} \frac{(tu/x_{i}; q)_{∞}}{(u/x_{i}; q)_{∞}} = ∑_{r≥0} u^{r}G_{r}(x; q, t).

In λ-ring notation (see Section 8), they can be written as

G_{r}(x; q, t) = h_{r} [\frac{1 − t}{1 − q} X^{†}] ,

with X^{†} = ∑_{i=1}^{n}(x_{i} + 1/x_{i}). Their specialization may be given as follows.

Lemma 2. For any positive integer r we have

G_{r}(t^{n−1}a, t^{n−2}a, …, a; q, t) = a^{r} \frac{(t^{n}; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, t^{n} \\ t^{−n}q^{1−r} \end{matrix}; q, qt^{1−2n}/a^{2}] .

Proof. Taking into account

∏_{i=1}^{n} \frac{(t^{n−i+1}ua; q)_{∞}}{(t^{n−i}ua; q)_{∞}} \frac{(t^{i−n+1}u/a; q)_{∞}}{(t^{i−n}u/a; q)_{∞}} = \frac{(t^{n}ua; q)_{∞}}{(ua; q)_{∞}} \frac{(tu/a; q)_{∞}}{(t^{1−n}u/a; q)_{∞}} ,

and applying the classical q–binomial formula [3, (II.3)]

\frac{(tu; q)_{∞}}{(u; q)_{∞}} = ∑_{i≥0} \frac{(t; q)_{i}}{(q; q)_{i}} u^{i},

we have

G_{r}(t^{n−1}a, t^{n−2}a, …, a; q, t) = ∑_{i=0}^{r} a^{(r−2i)}t^{i(1−n)} \frac{(t^{n}; q)_{i}}{(q; q)_{i}} \frac{(t^{n}; q)_{r−i}}{(q; q)_{r−i}} . □

Conjecture 1. For any positive integer r we have

g_{r}^{(C)}(q, t, T) = ∑_{i=0}^{[r/2]} G_{r−2i}(q, t) t^{i} \frac{(T/t; q)_{i}}{(q; q)_{i}} \frac{(t^{n}q^{r−i}; q)_{i}}{(T t^{n−1}q^{r−i}; q)_{i}} \frac{1 − t^{n}q^{r−2i}}{1 − t^{n}q^{r−i}} .

Conversely

G_{r}(q, t) = ∑_{i=0}^{[r/2]} g_{r−2i}^{(C)}(q, t, T) T^{i} \frac{(t/T; q)_{i}}{(q; q)_{i}} \frac{(t^{n}q^{r−2i}; q)_{i}}{(T t^{n−1}q^{r−2i+1}; q)_{i}} .

In other words, the transition matrix from g^{(C)}(q, t, T) to G(q, t) is M(t^{n}, t; T/t, 1), and its inverse is M(t^{n}, t; 1, T/t).

Using λ-ring techniques, we have proved this conjecture for T = t.

Theorem 4. For any positive integer r we have g_{r}^{(C)}(q, t, t) = G_{r}(q, t).

The proof will be given below in Section 9. The Specialization Formula gives some support to Conjecture 1.

Lemma 3. Conjecture 1 yields the specialization

g_{r}^{(C)}(t^{n−1}a, t^{n−2}a, …, a; q, t, T) = a^{r} \frac{(t^{n}; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, T t^{n−1} \\ t^{1−n}q^{1−r}/T \end{matrix} ; q, qt^{2−2n}/T a^{2}] .

Proof. A straightforward application of Theorem 1 and Lemma 2 with u = t^{n}, v = t/a^{2}, x = T/t, and y = 1. □

Corollary. Conjecture 1 is true for x_{i} = t^{n−i}T^{\frac{1}{2}} (1 ≤ i ≤ n).

Proof. Keeping the same notations, we now have v = 1/x. The previous result is a q–Chu–Vandermonde sum [3, (II.7)] given by

T^{r/2} \frac{(u; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, uv \\ q^{1−r}/uv \end{matrix}; q, q/u^{2}v^{2}] = T^{r/2} \frac{(u; q)_{r}}{(q; q)_{r}} \frac{(q^{1−r}/u^{2}v^{2}; q)_{r}}{(q^{1−r}/uv; q)_{r}}

= T^{r/2}(uv)^{−r} \frac{(u; q)_{r}}{(uv; q)_{r}} \frac{(u^{2}v^{2}; q)_{r}}{(q; q)_{r}} .

We recover the Specialization Formula. □

4. D versus C

The root system D_{n} is self-dual: R = R^{∨}. The set of positive roots is R^{+} = {ε_{i} ± ε_{j}, 1 ≤ i < j ≤ n}. Roots have all the same length and we write t_{α} = t for α ∈ R^{+}.

The Weyl group W is the semi-direct product of the permutation group S_{n} by (ℤ/2ℤ)^{n−1}. It acts on V by signed permutation of components, subject to the condition that the number of minus signs is even. The fundamental weights are given by ω_{i} = ∑_{j=1}^{i} ε_{j} for 1 ≤ i ≤ n − 2. The “spin weights” ω_{n−1} and ω_{n} are defined by ω_{n} = \frac{1}{2}(ε_{1} + … + ε_{n}) and ω_{n−1} = ω_{n} − ε_{n}. There are three minuscule weights ω_{1}, ω_{n−1} and ω_{n}.

The dominant weights λ ∈ P^{+} can be identified with vectors λ = ∑_{i=1}^{n} λ_{i}ε_{i}, whose components are all integers or all half-integers, and subject to the condition λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n−1} ≥ |λ_{n}|. The partial order λ ≥ μ is given by

∑_{i=1}^{j}(λ_{i} − μ_{i}) ∈ ℕ, for j = 1, …, n − 2,  ∑_{i=1}^{n−1}(λ_{i} − μ_{i}) ± (λ_{n} − μ_{n}) ∈ 2ℕ.

Writing t = q^{k} we have ρ_{k} = ρ_{k}^{∗} = k ∑_{i=1}^{n}(n − i)ε_{i}.

If we define

P_{rω_{1}}^{(D)}(x; q, t) = \frac{(q; q)_{r}}{(t; q)_{r}} g_{r}^{(D)}(x; q, t),

the Specialization Formula reads

g_{r}^{(D)}(t^{n−1}, t^{n−2}, …, 1; q, t) = t^{r(1−n)} \frac{(t^{n}; q)_{r}}{(q; q)_{r}} \frac{(t^{2n−2}; q)_{r}}{(t^{n−1}; q)_{r}} .

We have easily g_{r}^{(D)}(q, t) = g_{r}^{(C)}(q, t, 1) and Lemma 3 can be written as follows.

Lemma 4. Conjecture 1 yields the specialization

g_{r}^{(D)}(t^{n−1}a, t^{n−2}a, …, a; q, t) = a^{r} \frac{(t^{n}; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, t^{n−1} \\ t^{1−n}q^{1−r} \end{matrix}; q, t^{2−2n}q/a^{2}] .

As before, for a = 1 the previous expression is a q–Chu–Vandermonde sum, and we recover the Specialization Formula.

Conjecture 2. For any positive integer r we have

g_{r}^{(D)}(q, t) = ∑_{i=0}^{[r/2]} g_{r−2i}^{(C)}(q, t, T) T^{i} \frac{(1/T; q)_{i}}{(q; q)_{i}} \frac{(t^{n}q^{r−2i}; q)_{2i}}{(t^{n−1}q^{r−i}; q)_{i} (T t^{n−1}q^{r−2i+1}; q)_{i}} .

Conversely

g_{r}^{(C)}(q, t, T) = ∑_{i=0}^{[r/2]} g_{r−2i}^{(D)}(q, t) \frac{(T; q)_{i}}{(q; q)_{i}} \frac{(t^{n}q^{r−2i}; q)_{2i}}{(T t^{n−1}q^{r−i}; q)_{i} (t^{n−1}q^{r−2i+1}; q)_{i}} .

Namely, the transition matrix from g^{(D)}(q, t) to g^{(C)}(q, t, T) is M(t^{n}, t; 1/t, T/t) and its inverse is M(t^{n}, t; T/t, 1/t).

Conjectures 1 and 2 are consistent since the former, written for T = t and using Theorem 4, and the latter, written for T = 1, both yield

Conjecture 3. For any positive integer r we have

g_{r}^{(D)}(q, t) = ∑_{i=0}^{[r/2]} G_{r−2i}(q, t) t^{i} \frac{(1/t; q)_{i}}{(q; q)_{i}} \frac{(t^{n}q^{r−i}; q)_{i}}{(t^{n−1}q^{r−i}; q)_{i}} \frac{1 − t^{n}q^{r−2i}}{1 − t^{n}q^{r−i}} .

Conversely

G_{r}(q, t) = ∑_{i=0}^{[r/2]} g_{r−2i}^{(D)}(q, t) \frac{(t; q)_{i}}{(q; q)_{i}} \frac{(t^{n}q^{r−2i}; q)_{i}}{(t^{n−1}q^{r−2i+1}; q)_{i}} .

Equivalently, the transition matrix from g^{(D)}(q, t) to G(q, t) is M(t^{n}, t; 1/t, 1), and its inverse is M(t^{n}, t; 1, 1/t).

The corollary of Lemma 3 shows that Conjecture 3 is true for x_{i} = t^{n−i} (1 ≤ i ≤ n). Specialization yields another consistency argument.

Lemma 5. Assuming Lemmas 3 and 4, Conjecture 2 is true for x_{i} = t^{n−i}a, 1 ≤ i ≤ n.

Proof. A straightforward application of Theorem 1 with u = t^{n}, v = t/a^{2}, x = 1/t, and y = T/t. □

5. B versus D

For the type B root system, the set of positive roots is the union of R_{1} = {ε_{i} ± ε_{j}, 1 ≤ i < j ≤ n} and R_{2} = {ε_{i}, 1 ≤ i ≤ n}. Elements of each set have the same length, and we write t_{α} = t for α ∈ R_{1} and t_{α} = T for α ∈ R_{2}.

The Weyl group W is the semi-direct product of the permutation group S_{n} by (ℤ/2ℤ)^{n}. It acts on V by signed permutation of components. The fundamental weights are given by ω_{i} = ∑_{j=1}^{i} ε_{j} for 1 ≤ i ≤ n − 1, and ω_{n} = \frac{1}{2}(ε_{1} + … + ε_{n}). This is the only minuscule weight.

The dominant weights λ ∈ P^{+} can be identified with vectors λ = ∑_{i=1}^{n} λ_{i}ε_{i}, whose components are all integers or all half-integers, and subject to the condition λ_{1} ≥ λ_{2} ≥ … ≥ λ_{n} ≥ 0. The partial order λ ≥ μ is given by ∑_{i=1}^{j}(λ_{i} − μ_{i}) ∈ ℕ, for j = 1, …, n. Writing t = q^{k} and T = q^{K}, we have

ρ_{k} = ∑_{i=1}^{n} ((n − i)k + K/2) ε_{i}, ρ_{k}^{∗} = ∑_{i=1}^{n} ((n − i)k + K) ε_{i}.

If we define

P_{rω_{1}}^{(B)}(x; q, t, T) = \frac{(q; q)_{r}}{(t; q)_{r}} g_{r}^{(B)}(x; q, t, T),

the Specialization Formula reads

g_{r}^{(B)}(t^{n−1}T, t^{n−2}T, …, T; q, t, T) = \frac{t^{r(1−n)}}{T^{r}} \frac{(t^{n}; q)_{r}}{(q; q)_{r}} \frac{(T t^{2n−2}; q)_{r}}{(T t^{2n−2}; q)_{2r}} \frac{(T^{2}t^{2n−2}; q)_{2r}}{(T t^{n−1}; q)_{r}} .

Conjecture 4. For any positive integer r we have

g_{r}^{(B)}(q, t, T) = ∑_{i=0}^{r} g_{r−i}^{(D)}(q, t) \frac{(T; q)_{i}}{(q; q)_{i}} \frac{(t^{n}q^{r−i}; q)_{i}}{(t^{n−1}q^{r−i+1}; q)_{i}} \frac{(t^{2n−2}q^{2r−i+1}; q)_{i}}{(T t^{2n−2}q^{2r−i}; q)_{i}} .

Conversely

g_{r}^{(D)}(q, t) = ∑_{i=0}^{r} g_{r−i}^{(B)}(q, t, T) T^{i} \frac{(1/T; q)_{i}}{(q; q)_{i}} \frac{(t^{n}q^{r−i}; q)_{i}}{(t^{n−1}q^{r−i+1}; q)_{i}}

× \frac{(t^{2n−2}q^{2r−2i+1}; q)_{i}}{(T t^{2n−2}q^{2r−2i+1}; q)_{i}} \frac{1 − t^{2n−2}q^{2r}}{1 − t^{2n−2}q^{2r−i}} .

Namely, the transition matrix from g^{(B)}(q, t, T) to g^{(D)}(q, t) is N (t^{n}, t^{n−1}; T, 1), and its inverse is N (t^{n}, t^{n−1}; 1, T).

Of course for T = 1 we recover g_{r}^{(D)}(q, t) = g_{r}^{(B)}(q, t, 1).

Lemma 6. Conjectures 1 and 4 yield the specialization

g_{r}^{(B)}(t^{n−1}a, t^{n−2}a, …, a; q, t, T)

= (−1)^{r}t^{r(1−n)} \frac{(t^{n}; q)_{r}}{(q; q)_{r}} \frac{(T t^{2n−2}; q)_{r}}{(T t^{2n−2}; q)_{2r}} \frac{(T^{2}t^{2n−2}; q^{2})_{r}(t^{2n−2}q; q^{2})_{r}}{(T t^{n−1}; q)_{r}}

×_{4}ϕ_{3} [ \begin{matrix}q^{−r}, T t^{2n−2}q^{r}, −at^{n−1}, −1/a \\ t^{n−1}q^{\frac{1}{2}} , −t^{n−1}q^{\frac{1}{2}} , −T t^{n−1} \end{matrix} ; q, q] .

Proof. A straightforward application of Lemma 4 and Theorem 2 with u = t^{n}, v = t^{n−1} and x = T. □

Corollary. This property is true for a = T.

Proof. Keeping the notations u = t^{n}, v = t^{n−1} and x = T, we now have x = a. Applying the q–Saalschütz formula [3, (II.12)], we have

4ϕ_{3} [ \begin{matrix}q^{−r}, av^{2}q^{r}, −av, −1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −av \end{matrix}; q, q] = _{3}ϕ_{2} [ \begin{matrix}q^{−r}, av^{2}q^{r}, −1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} \end{matrix} ; q, q]

= \frac{(−avq^{\frac{1}{2}} q)_{r}}{(vq^{\frac{1}{2}} ; q)_{r}} \frac{(q^{\frac{1}{2}} q^{−r}/av; q)_{r}}{(−q^{\frac{1}{2}} q^{−r}/v; q)_{r}} .

This can be written

4ϕ_{3} [ \begin{matrix}q^{−r}, av^{2}q^{r}, −av, −1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −av \end{matrix}; q, q] = (−1/a)^{r} \frac{(avq^{\frac{1}{2}} ; q)_{r}}{(vq^{\frac{1}{2}} ; q)_{r}} \frac{(−avq^{\frac{1}{2}} ; q)_{r}}{(−vq^{\frac{1}{2}} ; q)_{r}}

= (−1/a)^{r} \frac{(a^{2}v^{2}; q)_{2r}}{(a^{2}v^{2}; q^{2})_{r}(v^{2}q; q^{2})_{r}} .

By substitution we recover the Specialization Formula. □

6. B versus C

The development of g_{r}^{(B)}(q, t, T) in terms of g_{r}^{(C)}(q, t, U) (and conversely) can be immediately written by composing the previous conjectures. We did not find a more compact expansion.

Conjecture 5. For any positive integer r we have

g_{r}^{(B)}(q, t, T) = ∑_{0≤i+2j≤r} g_{r−i−2j}^{(C)}(q, t, U) T^{j} \frac{(1/T; q)_{j}}{(q; q)_{j}} \frac{(U; q)_{i}}{(q; q)_{i}}

× \frac{(t^{n}q^{r−i−2j}; q)_{i+2j}}{(t^{n−1}q^{r−i−j}; q)_{i+j} (T t^{n−1}q^{r−i−2j+1}; q)_{j}} \frac{1 − t^{n−1}q^{r−i}}{1 − t^{n−1}q^{r}} \frac{(t^{2n−2}q^{2r−i+1}; q)_{i}}{(Ut^{2n−2}q^{2r−i}; q)_{i}} .

Conversely

g_{r}^{(C)}(q, t, T) = ∑_{0≤2i+j≤r} g_{r−2i−j}^{(B)}(q, t, U) U^{j} \frac{(1/U; q)_{j}}{(q; q)_{j}} \frac{(T; q)_{i}}{(q; q)_{i}}

\frac{(t^{2n−2}q^{2r−4i−2j+1}; q)_{j}}{(Ut^{2n−2}q^{2r−4i−2j+1}; q)_{j}} \frac{1 − t^{2n−2}q^{2r−4i}}{1 − t^{2n−2}q^{2r−4i−j}} \frac{(t^{n}q^{r−2i−j}; q)_{2i+j}}{(T t^{n−1}q^{r−i}; q)_{i}(t^{n−1}q^{r−2i−j+1}; q)_{i+j}} .

7. Open problems

The statements of Lemmas 3, 4 and 6 can be written in a rather similar form by applying the following property.

Lemma 7. We have

2ϕ_{1} [ \begin{matrix}q^{−r}, v \\ q^{1−r}/v \end{matrix}; q, q/v^{2}a^{2}] = (av)^{−r} \frac{(v^{2}; q)_{r}}{(v; q)_{r}} _{4}ϕ_{3} [ \begin{matrix}q^{−r}, v^{2}q^{r}, av, 1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −v \end{matrix}; q, q] .

Proof. A straightforward consequence of [3, (7.4.12) and (7.4.13)]. □

Writing g_{r}^{(R)}(a; q, t, T) for g_{r}^{(R)}(t^{n−1}a, t^{n−2}a, …, a; q, t, T), we then have

g_{r}^{(C)}(a; q, t, T) = (−1)^{r}g_{r}^{(C)}(T^{\frac{1}{2}} ; q, t, T)
×_{4}ϕ_{3} [ \begin{matrix}q^{−r}, T^{2}t^{2n−2}q^{r}, −aT^{\frac{1}{2}} t^{n−1}, −T^{\frac{1}{2}} /a \\ T t^{n−1}q^{\frac{1}{2}} , −T t^{n−1}q^{\frac{1}{2}} , −T t^{n−1} \end{matrix} ; q, q]
g_{r}^{(D)}(a; q, t) = (−1)^{r}g_{r}^{(D)}(1; q, t)
×_{4}ϕ_{3} [ \begin{matrix}q^{−r}, t^{2n−2}q^{r}, −at^{n−1}, −1/a \\ t^{n−1}q^{\frac{1}{2}} , −t^{n−1}q^{\frac{1}{2}} , −t^{n−1} \end{matrix} ; q, q]
g_{r}^{(B)}(a; q, t, T) = (−T)^{r}g_{r}^{(B)}(T; q, t, T)\frac{(t^{2n−2}q; q^{2})_{r}}{(T^{2}t^{2n−2}q; q^{2})_{r}}
×_{4}ϕ_{3} [ \begin{matrix}q^{−r}, T t^{2n−2}q^{r}, −at^{n−1}, −1/a \\ t^{n−1}q^{\frac{1}{2}} , −t^{n−1}q^{\frac{1}{2}} , −T t^{n−1} \end{matrix} ; q, q] .

These formulas seem difficult to unify in a general conjecture written in terms of the root system R.

We are also in lack of a conjecture for the generating function of g_{r}^{(R)}, except when R = C and T = t (see Theorem 4).

In another paper, we shall present conjectures giving for R ∈ {B, C, D},

(i) a “generalized Pieri formula” expanding g_{r}^{(R)}g_{s}^{(R)} in terms of the Macdonald polynomials P_{λ_{1}ω_{1}+λ_{2}ω_{2}}^{(R)},

(ii) conversely, the expansion of any Macdonald polynomial P_{λ_{1}ω_{1}+λ_{2}ω_{2}}^{(R)} in terms of products g_{r}^{(R)}g_{s}^{(R)} (“inverse Pieri formula”).

8. λ-rings

This section and the following will be devoted to the proof of Theorem 4. This will be done in the language of λ-rings, which turns out to be the most efficient. Here we only intend to give a short survey of this theory. Details and other applications may be found, for instance, in [6, 7], and in some examples of [10] (see pp. 25, 43, 65 and 79).

The basic idea of the theory of λ-rings is the following. A symmetric function f is usually understood as evaluated on a set of variables A = {a_{1}, a_{2}, a_{3}, … }, the value being denoted f(A). When using λ-rings, this interpretation is not the main one. Symmetric functions are first understood as operators on polynomials. Thus any symmetric function f is first understood as acting on the polynomial P, mapping P to f[P]. Of course the standard interpretation may be recovered as a special case. These statements may be made more precise as follows.

Let A = {a_{1}, a_{2}, a_{3}, … } be a (finite or infinite) set of independent indeterminates, called an alphabet. We introduce the generating functions

E_{u}(A) = ∏_{a∈A}(1 + ua), H_{u}(A) = ∏_{a∈A} \frac{1}{1 − ua}, P_{u}(A) = ∑_{a∈A} \frac{a}{1 − ua},

whose development defines symmetric functions known as elementary functions e_{k}(A), complete functions h_{k}(A), and power sums p_{k}(A), respectively. Each of these three sets generate the symmetric algebra 𝕊(A).

We define an action, denoted [ ], of 𝕊(A) on the ring ℝ[A] of polynomials in A with real coefficients. Since the power sums p_{k} algebraically generate 𝕊(A), it is enough to define the action of p_{k} on ℝ[A]. Writing any polynomial as ∑_{c,P} cP, with c a real constant and P a monomial in (a_{1}, a_{2}, a_{3}, …), we define

p_{k} [∑_{c,P} cP ] = ∑_{c,P} cP^{k}.

This action extends to 𝕊[A]. For instance we obtain

E_{u} [∑_{c,P} cP ] = ∏_{c,P}(1 + uP)^{c} , H_{u} [∑_{c,P} cP ] = ∏_{c,P}(1 − uP)^{−c}.

More generally, we can define an action of 𝕊(A) on the ring of rational functions, and even on the ring of formal series, by writing

p_{k} (\frac{∑ cP}{∑ dQ}) = \frac{∑ cP^{k}}{∑ dQ^{k}}

with c, d real constants and P, Q monomials in (a_{1}, a_{2}, a_{3}, …). This action still extends to 𝕊(A).

As an example, being given a (finite or infinite) alphabet A = {a_{1}, a_{2}, a_{3}, … }, let us compute H_{u}[h_{1}(A)] and H_{u}[h_{2}(A)]. We obtain [10, Example 1.5.10, p. 79]

H_{u}[h_{1}(A)] = ∏_{i}(1 − ua_{i})^{−1},
H_{u}[h_{2}(A)] = ∏_{i}(1 − ua_{i}^{2})^{−1} ∏_{i<j}(1 − ua_{i}a_{j})^{−1}.

In the following we shall write

A^{†} = h_{1}(A) = ∑_{i} a_{i}.

Bu definition we have p_{k}[A^{†}] = ∑_{i} a_{i}^{k}. Thus p_{k}[A^{†}] = p_{k}(A), which yields that for any symmetric function f, we have f[A^{†}] = f(A). In particular

f(1, q, q^{2}, q^{3}, …, q^{m−1}) = f [\frac{1 − q^{m}}{1 − q} ] , f(1, q, q^{2}, q^{3}, …) = f [\frac{1}{1 − q} ] .

The following relations are straightforward consequences of the previous definitions. For any formal series P, Q, we have

h_{r}[P + Q] = ∑_{k=0}^{r} h_{r−k}[P] h_{k}[Q], e_{r}[P + Q] = ∑_{k=0}^{r} e_{r−k}[P] e_{k}[Q].

Or equivalently

H_{u}[P + Q] = H_{u}[P] H_{u}[Q], E_{u}[P + Q] = E_{u}[P] E_{u}[Q]
H_{u}[P − Q] = H_{u}[P] H_{u}[Q]^{−1}, E_{u}[P − Q] = E_{u}[P] E_{u}[Q]^{−1}.

As an application, for a finite alphabet A = {a_{1}, a_{2}, …, a_{m}} we may write

H_{1} [\frac{uA^{†}}{1 − q} ] = ∏_{i≥0} H_{1}[uq^{i}A^{†}] = ∏_{k=1}^{m} ∏_{i≥0} H_{1}[uq^{i}a_{k}]

= ∏_{k=1}^{m} ∏_{i≥0} \frac{1}{1 − uq^{i}a_{k}} = ∏_{k=1}^{m} \frac{1}{(ua_{k}; q)_{∞}} ,

and

H_{1} [\frac{1 − t}{1 − q} A^{†}] = H_{1} [\frac{A^{†}}{1 − q} ] (H_{1} [\frac{tA^{†}}{1 − q} ])^{−1}.

Finally we obtain

H_{1} [\frac{1 − t}{1 − q} A^{†}] = ∑_{r≥0} h_{r} [\frac{1 − t}{1 − q} A^{†}] = ∏_{i=1}^{m} \begin{matrix}(ta_{i}; q) & ∞ \\ (a_{i}; q) & ∞ \end{matrix} .

9. Proof of Theorem 4

In this section we assume that R is of type C_{n} with T = t. We define x_{i} = e^{ε_{i}} for i = 1 … n, and regard elements of A as Laurent polynomials of n variables x_{1}, …, x_{n}.

The dual root system R^{∨} = B_{n} has one minuscule weight π = \frac{1}{2}(ε_{1} + … + ε_{n}). With the notations of Section 1, we have

Φ_{π} = ∏_{i=1}^{n} \frac{1 − tx_{i}^{2}}{1 − x_{i}^{2}} ∏_{1≤i<j≤n} \frac{1 − tx_{i}x_{j}}{1 − x_{i}x_{j}} ,

and the translation operator T_{π} acts on A by

T_{π}f(x_{1}, …, x_{n}) = f(q^{\frac{1}{2}} x_{1}, …, q^{\frac{1}{2}} x_{n}).

The Weyl group W is the semi-direct product of the permutation group S_{n} by (ℤ/2ℤ)^{n}. It acts on V by signed permutation of components. Hence the W -orbit of π is formed by vectors \frac{1}{2}(σ_{1}ε_{1} + … + σ_{n}ε_{n}) with σ ∈ (−1, +1)^{n}.

The Macdonald operator E_{π} can be written as

E_{π}f = ∑_{σ∈(−1,+1)^{n}} ∏_{i=1}^{n} \frac{1 − tx_{i}^{2σ_{i}}}{1 − x_{i}^{2σ_{i}}} ∏_{1≤i<j≤n} \frac{1 − tx_{i}^{σ_{i}}x_{j}^{σ_{j}}}{1 − x_{i}^{σ_{i}}x_{j}^{σ_{j}}} f(q^{σ_{1}/2}x_{1}, …, q^{σ_{n}/2}x_{n}).

Up to a constant, the Macdonald polynomial P_{rω_{1}} is defined by

E_{π}P_{rω_{1}} = ∏_{i=1}^{n−1}(t^{i} + 1) (t^{n}q^{r/2} + q^{−r/2}) P_{rω_{1}}.

This normalization constant is given by the condition

P_{rω_{1}} = m_{rω_{1}} + lower terms,

where the orbit-sum m_{rω_{1}} is given by m_{rω_{1}}(x) = ∑_{i=1}^{n}(x_{i}^{r} + 1/x_{i}^{r}).

The following notations will be used till the end of this paper. We write

X_{+} = {x_{1}, …, x_{n}}, X_{−} = {1/x_{1}, …, 1/x_{n}}, X = X_{+} ∪ X_{−},

so that X^{†} = ∑_{i=1}^{n}(x_{i} + 1/x_{i}). Observe that Φ_{π} = H_{1}[(1 − t)h_{2}(X_{+})]. We consider the generating series

H_{1} [\frac{1 − t}{1 − q} X^{†}] = ∑_{r≥0} u^{r} h_{r} [\frac{1 − t}{1 − q} X^{†}] = ∏_{i=1}^{n} \frac{(tux_{i}; q)_{∞}}{(ux_{i}; q)_{∞}} \frac{(tu/x_{i}; q)_{∞}}{(u/x_{i}; q)_{∞}} .

Theorem 4 states that

g_{r}^{(C)}(x; q, t, t) = h_{r} [\frac{1 − t}{1 − q} X^{†}] .

Since it is well known ([10, p. 314], [7, p. 237]) that for any alphabet A = {a_{1}, …, a_{m}} one has

h_{r} [\frac{1 − t}{1 − q} A^{†}] = \frac{(t; q)_{r}}{(q; q)_{r}} ∑_{i=1}^{m} a_{i}^{r} + other terms,

we have only to prove

E_{π} h_{r} [\frac{1 − t}{1 − q} X^{†}] = ∏_{i=1}^{n−1}(t^{i} + 1) (t^{n}q^{r/2} + q^{−r/2}) h_{r} [\frac{1 − t}{1 − q} X^{†}] .

This will be established under the following equivalent form.

Theorem 4’. We have

E_{π}H_{1} [\frac{1 − t}{1 − q} X^{†}] = ∏_{i=1}^{n−1}(t^{i} + 1) (t^{n}H_{1} [\frac{1 − t}{1 − q} q^{\frac{1}{2}} X^{†}] + H_{1} [\frac{1 − t}{1 − q} q^{− \frac{1}{2}} X^{†}]) .

In order to evaluate the left-hand side of this identity, we shall need the following trick. For σ = ±1 we have

xq^{σ/2} + \frac{1}{xq^{σ/2}} = q^{\frac{1}{2}} (x + \frac{1}{x}) + q^{− \frac{1}{2}} (1 − q)x^{−σ},

which is checked separately for σ = 1 and σ = −1. Consequently we write

H_{1} [\frac{1 − t}{1 − q} ∑_{i=1}^{n} (x_{i}q^{σ_{i}/2} + \frac{1}{x_{i}q^{σ_{i}/2}} )] = H_{1} [\frac{1 − t}{1 − q} q^{\frac{1}{2}} X^{†} + (1 − t)q^{− \frac{1}{2}} ∑_{i=1}^{n} x_{i}^{−σ_{i}}]

= H_{1} [(1 − t)q^{− \frac{1}{2}} ∑_{i=1}^{n} x_{i}^{−σ_{i}}] H_{1} [\frac{1 − t}{1 − q} q^{\frac{1}{2}} X^{†}]

= ∏_{i=1}^{n} \frac{1 − tq^{− \frac{1}{2}} x_{i}^{−σ_{i}}}{1 − q^{− \frac{1}{2}} x_{i}^{−σ_{i}}} H_{1} [\frac{1 − t}{1 − q} q^{\frac{1}{2}} X^{†}] .

Similarly, on the right-hand side we get

H_{1} [\frac{1 − t}{1 − q} q^{− \frac{1}{2}} X^{†}] = H_{1} [\frac{1 − t}{1 − q} q^{\frac{1}{2}} X^{†} + (1 − t)q^{− \frac{1}{2}} X^{†}]

= H_{1}[(1 − t)q^{− \frac{1}{2}} X^{†}] H_{1} [\frac{1 − t}{1 − q} q^{\frac{1}{2}} X^{†}]

= ∏_{i=1}^{n} \frac{1 − tq^{− \frac{1}{2}} x_{i}}{1 − q^{− \frac{1}{2}} x_{i}} \frac{1 − tq^{− \frac{1}{2}} /x_{i}}{1 − q^{− \frac{1}{2}} /x_{i}} H_{1} [\frac{1 − t}{1 − q} q^{\frac{1}{2}} X^{†}] .

By substitution, we see that the statement of Theorem 4′ is equivalent to the following rational identity, written with u = q^{− \frac{1}{2}} ,

∑_{σ∈(−1,+1)^{n}} ∏_{i=1}^{n} \frac{1 − tx_{i}^{2σ_{i}}}{1 − x_{i}^{2σ_{i}}} \frac{1 − tux_{i}^{−σ_{i}}}{1 − ux_{i}^{−σ_{i}}} ∏_{1≤i<j≤n} \frac{1 − tx_{i}^{σ_{i}}x_{j}^{σ_{j}}}{1 − x_{i}^{σ_{i}}x_{j}^{σ_{j}}} =

∏_{i=1}^{n−1}(t^{i} + 1) (t^{n} + ∏_{i=1}^{n} \frac{1 − tux_{i}}{1 − ux_{i}} \frac{1 − tu/x_{i}}{1 − u/x_{i}} ) .

Writing T_{i} for the operator x_{i} → 1/x_{i}, we are led to prove Theorem 4 under the following equivalent form.

Theorem 5. We have

(1 + T_{1}) ⋯ (1 + T_{n}) (∏_{i=1}^{n} \frac{1 − tx_{i}^{2}}{1 − x_{i}^{2}} \frac{1 − tu/x_{i}}{1 − u/x_{i}} ∏_{1≤i<j≤n} \frac{1 − tx_{i}x_{j}}{1 − x_{i}x_{j}} ) =

∏_{i=1}^{n−1}(t^{i} + 1) (t^{n} + ∏_{i=1}^{n} \frac{1 − tux_{i}}{1 − ux_{i}} \frac{1 − tu/x_{i}}{1 − u/x_{i}} ) .

Both sides of this identity are rational functions of u having poles at u = x_{i} and u = 1/x_{i}, for i = 1, …, n. We first prove that their constant terms are equal, i.e. that the statement is true for u = 0.

Lemma 8. We have

(1 + T_{1}) ⋯ (1 + T_{n}) Φ_{π} = ∏_{i=1}^{n}(t^{i} + 1)

Proof. This is a direct consequence of Weyl’s denominator formula

δ = e^{−ρ} ∏_{α∈R^{+}}(e^{α} − 1) = ∑_{w∈W} det(w) e^{wρ},

with ρ = ∑_{i=1}^{n}(n − i + 1)ε_{i}, from which follows

Φ_{π} = δ^{−1} ∑_{w∈W} det(w) t^{⟨π,ρ+wρ⟩}e^{wρ}.

Since for any w ∈ W, we have wδ = det(w)δ, we obtain

(1 + T_{1}) ⋯ (1 + T_{n}) Φ_{π} = ∑_{τ∈W (π)} t^{⟨π+τ,ρ⟩} = ∑_{σ∈(−1,+1)^{n}} ∏_{i=1}^{n} t^{\frac{1}{2} (n−i+1)(1+σ_{i})}. □

Proof of Theorem 5. It is sufficient to prove that both sides of the identity have the same residue at each of their poles, i.e. at u = x_{i} and u = 1/x_{i}, 1 ≤ i ≤ n. By symmetry, this has only to be checked for some x_{i}, say x_{n}. We shall only do it at u = x_{n}, the proof at u = 1/x_{n} being similar.

If A = {a_{1}, …, a_{m}} is an arbitrary alphabet, we have

∏_{i=1}^{m} \frac{tu − a_{i}}{u − a_{i}} = t^{m} + (t − 1) ∑_{i=1}^{m} \frac{a_{i}}{u − a_{i}} _{∏_{j=1}^{m} ; j≠i} \frac{ta_{i} − a_{j}}{a_{i} − a_{j}} .

This decomposition as a sum of partial fractions is actually a Lagrange interpolation (see [5], and also [7, p. 236]). We first apply it to the right-hand side of the identity. Its residue at u = x_{n} is given by

∏_{i=1}^{n−1}(t^{i} + 1) x_{n} (t − 1) ∏_{j=1}^{n−1} \frac{tx_{n} − x_{j}}{x_{n} − x_{j}} ∏_{j=1}^{n} \frac{tx_{n} − 1/x_{j}}{x_{n} − 1/x_{j}} .

We then apply the Lagrange interpolation to

∏_{i=1}^{n} \frac{tu − x_{i}^{σ_{i}}}{u − x_{i}^{σ_{i}}} ,

on the left-hand side of the identity. Only fractions with σ_{n} = 1 contribute to the residue at u = x_{n}, which can be written as

x_{n}(t − 1)\frac{1 − tx_{n}^{2}}{1 − x_{n}^{2}}

× ∑_{σ∈(−1,+1)^{n−1}} ∏_{i=1}^{n−1} \frac{1 − tx_{i}^{2σ_{i}}}{1 − x_{i}^{2σ_{i}}} \frac{1 − tx_{n}x_{i}^{−σ_{i}}}{1 − x_{n}x_{i}^{−σ_{i}}} \frac{1 − tx_{i}^{σ_{i}}x_{n}}{1 − x_{i}^{σ_{i}}x_{n}} ∏_{1≤i<j≤n−1} \frac{1 − tx_{i}^{σ_{i}}x_{j}^{σ_{j}}}{1 − x_{i}^{σ_{i}}x_{j}^{σ_{j}}} .

By identification of residues on both sides, and using an obvious induction on n, we are led to prove the identity

(1 + T_{1}) ⋯ (1 + T_{n}) (∏_{i=1}^{n} \frac{1 − tx_{i}^{2}}{1 − x_{i}^{2}} \frac{1 − tux_{i}}{1 − ux_{i}} \frac{1 − tu/x_{i}}{1 − u/x_{i}} ∏_{1≤i<j≤n} \frac{1 − tx_{i}x_{j}}{1 − x_{i}x_{j}} ) =

∏_{i=1}^{n}(t^{i} + 1) \frac{1 − tux_{i}}{1 − ux_{i}} \frac{1 − tu/x_{i}}{1 − u/x_{i}} .

Since

∏_{i=1}^{n} \frac{1 − tux_{i}}{1 − ux_{i}} \frac{1 − tu/x_{i}}{1 − u/x_{i}} = H_{1}[(1 − t)uX^{†}]

is obviously invariant under any T_{i}, the left-hand side may be written

∏_{i=1}^{n} \frac{1 − tux_{i}}{1 − ux_{i}} \frac{1 − tu/x_{i}}{1 − u/x_{i}} (1 + T_{1}) ⋯ (1 + T_{n})Φ_{π}.

We conclude by applying Lemma 8. □

20 MICHEL LASSALLE

10. Proofs of Theorems 1, 2 and 3

We shall implicitly use many of the formulas about q-shifted factorials, listed in Appendix I of [3]. In particular we shall write

(a_{1}, a_{2}, …, a_{r}; q)_{i} = (a_{1}; q)_{i} (a_{2}; q)_{i} … (a_{r}; q)_{i}.

Theorem 1. We have the following transformation between _{2}ϕ_{1} series

\frac{(u; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, ux \\ q^{1−r}/ux \end{matrix}; q, qv/u^{2}x]

= ∑_{i=0}^{[r/2]} \frac{(u; q)_{r−2i}}{(q; q)_{r−2i}} _{2}ϕ_{1} [ \begin{matrix}q^{2i−r}, uy \\ q^{1+2i−r}/uy \end{matrix}; q, qv/u^{2}y] y^{i}v^{i} \frac{(x/y; q)_{i} (uq^{r−2i}; q)_{2i}}{(q, uxq^{r−i}, uyq^{r−2i+1}; q)_{i}} .

Proof [14]. With a = v/u, applying [3, (3.4.7)], we have

\frac{(u; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, ux \\ q^{1−r}/ux \end{matrix}; q, qv/u^{2}x] = a^{r} \frac{(u, 1/a^{2}; q)_{r}}{(q, 1/a; q)_{r}}

×_{8}ϕ_{7} [ \begin{matrix}q^{−r}a, q(q^{−r}a)^{\frac{1}{2}} , −q(q^{−r}a)^{\frac{1}{2}} , aux, q^{− \frac{r}{2}} , −q^{− \frac{r}{2}} , q^{\frac{1}{2} − \frac{r}{2}} , −q^{\frac{1}{2} − \frac{r}{2}} \\ (q^{−r}a)^{\frac{1}{2}} , −(q^{−r}a)^{\frac{1}{2}} , q^{1−r}/ux, q^{1− \frac{r}{2}} a, −q^{1− \frac{r}{2}} a, q^{\frac{1}{2} − \frac{r}{2}} a, −q^{\frac{1}{2} − \frac{r}{2}} a \end{matrix}; q, qa/ux] ,

or equivalently

\frac{(u; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, ux \\ q^{1−r}/ux \end{matrix}; q, qv/u^{2}x] = a^{r} \frac{(u, 1/a^{2}; q)_{r}}{(q, 1/a; q)_{r}}

× ∑_{k=0}^{[r/2]} \frac{1 − aq^{2k−r}}{1 − aq^{−r}} \frac{(q^{−r}a, aux; q)_{k}}{(q^{1−r}/ux, q; q)_{k}} \frac{(q^{−r}; q)_{2k}}{(q^{1−r}a^{2}; q)_{2k}} (qa/ux)^{k}.

This yields

\frac{(u; q)_{r−2i}}{(q; q)_{r−2i}} _{2}ϕ_{1} [ \begin{matrix}q^{2i−r}, uy \\ q^{1+2i−r}/uy \end{matrix}; q, qv/u^{2}y]

= (v/u)^{r} \frac{(u, u^{2}/v^{2}; q)_{r}}{(q, u/v; q)_{r}} (q/u)^{2i} \frac{(q^{−r}, q^{1−r}v/u; q)_{2i}}{(q^{1−r}/u, q^{1−r}v^{2}/u^{2}; q)_{2i}}

∑_{k=0}^{[r/2]−i} \frac{1 − q^{2i+2k−r}v/u}{1 − q^{2i−r}v/u} \frac{(q^{2i−r}v/u, vy; q)_{k}}{(q^{1+2i−r}/uy, q; q)_{k}} \frac{(q^{2i−r}; q)_{2k}}{(q^{1+2i−r}v^{2}/u^{2}; q)_{2k}} (qv/u^{2}y)^{k}.

If we substitute this value in the right-hand side of the identity, and put k = j − i, we obtain

(v/u)^{r} \frac{(u, u^{2}/v^{2}; q)_{r}}{(q, u/v; q)_{r}} ∑_{j=0}^{[r/2]} ∑_{i=0}^{j} (q/u)^{2i} \frac{(q^{−r}, q^{1−r}v/u; q)_{2i}}{(q^{1−r}/u, q^{1−r}v^{2}/u^{2}q)_{2i}}

× \frac{1 − q^{2j−r}v/u}{1 − q^{2i−r}v/u} \frac{(q^{2i−r}v/u, vy; q)_{j−i}}{(q^{1+2i−r}/uy, q; q)_{j−i}} \frac{(q^{2i−r}; q)_{2j−2i}}{(q^{1+2i−r}v^{2}/u^{2}; q)_{2j−2i}} (qv/u^{2}y)^{j−i}

× y^{i}v^{i} \frac{(x/y; q)_{i} (uq^{r−2i}; q)_{2i}}{(q, uxq^{r−i}, uyq^{r−2i+1}; q)_{i}} .

MACDONALD POLYNOMIALS 21

This may be written as

(v/u)^{r} \frac{(u, u^{2}/v^{2}; q)_{r}}{(q, u/v; q)_{r}}

× ∑_{j=0}^{[r/2]} \frac{1 − q^{2j−r}v/u}{1 − q^{−r}v/u} \frac{(q^{−r}v/u, vy; q)_{j}}{(q^{1−r}/uy, q; q)_{j}} \frac{(q^{−r}; q)_{2j}}{(q^{1−r}v^{2}/u^{2}; q)_{2j}} (qv/u^{2}y)^{j}

× ∑_{i=0}^{j} \frac{(q^{1−r}/uy; q)_{2i}}{(q^{1−r}/u; q)_{2i}} \frac{(q^{−j}, q^{j−r}v/u, x/y; q)_{i} (uq^{r−2i}; q)_{2i}}{(q^{1−j}/vy, q^{1+j−r}/uy, q, uxq^{r−i}, uyq^{r−2i+1}; q)_{i}} (q^{2}/vy)^{i}.

The sum over i reads

∑_{i=0}^{j} \frac{1 − uyq^{r−2i}}{1 − uyq^{r−i}} \frac{(q^{−j}, q^{j−r}v/u, q^{1−r}/uy, x/y; q)_{i}}{(q^{1−j}/vy, q^{1+j−r}/uy, q^{1−r}/ux, q; q)_{i}} (q^{2}/vx)^{i}

= _{6}ϕ_{5} [ \begin{matrix}q^{−r}/uy, q(q^{−r}/uy)^{\frac{1}{2}} , −q(q^{−r}/uy)^{\frac{1}{2}} , x/y, q^{j−r}v/u, q^{−j} \\ (q^{−r}/uy)^{\frac{1}{2}} , −(q^{−r}/uy)^{\frac{1}{2}} , q^{1−r}/ux, q^{1−j}/vy, q^{j+1−r}/uy, \end{matrix}; q, q/vx] .

By [3, (II.21)] this terminating very-well-poised _{6}ϕ_{5} sum equals

\frac{(q^{1−r}/uy, q^{1−j}/vx; q)_{j}}{(q^{1−r}/ux, q^{1−j}/vy; q)_{j}} = \frac{(q^{1−r}/uy, vx; q)_{j}}{(q^{1−r}/ux, vy; q)_{j}} (y/x)^{j}.

Finally we have proved that the right-hand side of the identity is

(v/u)^{r} \frac{(u, u^{2}/v^{2}; q)_{r}}{(q, u/v; q)_{r}}

× ∑_{j=0}^{[r/2]} \frac{1 − q^{2j−r}v/u}{1 − q^{−r}v/u} \frac{(q^{−r}v/u, vx; q)_{j}}{(q^{1−r}/ux, q; q)_{j}} \frac{(q^{−r}; q)_{2j}}{(q^{1−r}v^{2}/u^{2}; q)_{2j}} (qv/u^{2}x)^{j}.

Hence the statement. □ Lemma 9. We have

4ϕ_{3} [ \begin{matrix}q^{−n}, a^{2}, qa, b \\ a, qa^{2}/b, q^{1+n}a^{2} \end{matrix} ; q, q^{1+n}a/b]

= \frac{(qa^{2}, −1; q)_{n}}{(q^{\frac{1}{2}} a, −q^{\frac{1}{2}} a; q)_{n}} _{4}ϕ_{3} [ \begin{matrix}q^{−n}, −qa/b, q^{\frac{1}{2}} a, −q^{\frac{1}{2}} a \\ −q^{1−n}, qa^{2}/b, −qa \end{matrix}; q, q] .

Proof [14]. The left-hand side obviously equals

8ϕ_{7} [ \begin{matrix}a^{2}, qa, −qa, b, −a, q^{\frac{1}{2}} a, −q^{\frac{1}{2}} a, q^{−n} \\ a, −a, qa^{2}/b, −qa, q^{\frac{1}{2}} a, −q^{\frac{1}{2}} a, q^{1+n}a^{2} \end{matrix} ; q, q^{1+n}a/b] .

By applying Watson’s transformation formula [3, (III.19)], it can be transformed into the right-hand side. □

22 MICHEL LASSALLE

Theorem 2. Defining

\frac{(u; q)_{r}}{(q; q)_{r}} H_{r} = ∑_{i=0}^{r} a^{r−i} \frac{(u; q)_{r−i}}{(q; q)_{r−i}} _{2}ϕ_{1} [ \begin{matrix}q^{i−r}, v \\ q^{1+i−r}/v \end{matrix}; q, q/v^{2}a^{2}] \frac{(x, uq^{r−i}, v^{2}q^{2r−i+1}; q)_{i}}{(q, vq^{r−i+1}, xv^{2}q^{2r−i}; q)_{i}} ,

we have

H_{r} = (−1/v)^{r} \frac{(vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −xv, xv^{2}; q)_{r}}{(xv^{2}; q)_{2r}} _{4}ϕ_{3} [ \begin{matrix}q^{−r}, xv^{2}q^{r}, −av, −1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −xv \end{matrix}; q, q] .

Proof [14]. Applying Lemma 7 we obtain

H_{r} = \frac{(q; q)_{r}}{(v; q)_{r}} ∑_{i=0}^{r} \frac{1 − vq^{r−i}}{1 − vq^{r}} \frac{(v^{2}; q)_{r−i}}{(q; q)_{r−i}} \frac{(x, v^{2}q^{2r−i+1}; q)_{i}}{(q, xv^{2}q^{2r−i}; q)_{i}} v^{i−r}

×_{4}ϕ_{3} [ \begin{matrix}q^{i−r}, v^{2}q^{r−i}, av, 1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −v \end{matrix}; q, q] .

Reversing the order of summation, this may be rewritten as

H_{r} = v^{−r} \frac{(v^{2}; q)_{r}}{(v; q)_{r}} ∑_{i=0}^{r} \frac{(q^{−r}, v^{2}q^{r}, av, 1/a; q)_{i}}{(q, vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −v; q)_{i}} q^{i}

×_{4}ϕ_{3} [ \begin{matrix}q^{−2r}/v^{2}, q^{1−r}/v, x, q^{i−r} \\ q^{−r}/v, q^{1−2r}/xv^{2}, q^{1−r−i}/v^{2} \end{matrix} ; q, q^{1−i}/xv] .

Using Lemma 9 with a := q^{−r}/v, b := x and n := r − i, the _{4}ϕ_{3} series equals

\frac{(q^{1−2r}/v^{2}, −1; q)_{r}}{(q^{\frac{1}{2} −r}/v, −q^{\frac{1}{2} −r}/v; q)_{r}} \frac{(vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} ; q)_{i}}{(v^{2}q^{r}, −q^{1−r}; q)_{i}}

×_{4}ϕ_{3} [ \begin{matrix}q^{i−r}, q^{\frac{1}{2} −r}/v, −q^{\frac{1}{2} −r}/v, −q^{1−r}/xv \\ q^{1−2r}/xv^{2}, −q^{1−r}/v, −q^{1−r+i} \end{matrix}; q, q] ,

which yields

H_{r} = \frac{(v^{2}, v^{2}q^{r}, −1; q)_{r}}{(v, vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} ; q)_{r}} v^{−r} q^{−\binom{r}{2}} ∑_{i=0}^{r} q^{i} \frac{(q^{−r}, av, 1/a; q)_{i}}{(q, −v, −q^{1−r}; q)_{i}}

×_{4}ϕ_{3} [ \begin{matrix}q^{i−r}, q^{\frac{1}{2} −r}/v, −q^{\frac{1}{2} −r}/v, −q^{1−r}/xv \\ q^{1−2r}/xv^{2}, −q^{1−r}/v, −q^{1−r+i} \end{matrix}; q, q] .

Reversing again the order of summation, this may be rewritten as

H_{r} = (−v, −1; q)_{r} v^{−r} q^{−\binom{r}{2}} ∑_{i=0}^{r} q^{i} \frac{(q^{−r}, q^{\frac{1}{2} −r}/v, −q^{\frac{1}{2} −r}/v, −q^{1−r}/xv; q)_{i}}{(q, q^{1−2r}/xv^{2}, −q^{1−r}/v, −q^{1−r}; q)_{i}}

×_{3}ϕ_{2} [ \begin{matrix}q^{i−r}, av, 1/a \\ −v, −q^{1−r+i} \end{matrix}; q, q] .

MACDONALD POLYNOMIALS 23

Since it is a q-Saalschütz sum [3, (II.12)], the _{3}ϕ_{2} series equals

\frac{(−av, −1/a; q)_{r−i}}{(−v, −1; q)_{r−i}} = \frac{(−av, −1/a; q)_{r}}{(−v, −1; q)_{r}} \frac{(−q^{1−r}, −q^{1−r}/v; q)_{i}}{(−q^{1−r}/av, −aq^{1−r}; q)_{i}} .

Finally we obtain

H_{r} = (−av, −1/a; q)_{r} v^{−r} q^{−\binom{r}{2}} _{4}ϕ_{3} [ \begin{matrix}q^{−r}, q^{\frac{1}{2} −r}/v, −q^{\frac{1}{2} −r}/v, −q^{1−r}/xv \\ q^{1−2r}/xv^{2}, −q^{1−r}/av, −aq^{1−r} \end{matrix}; q, q] ,

and we conclude easily. □ Theorem 3. We have

a^{r} \frac{(u; q)_{r}}{(q; q)_{r}} _{2}ϕ_{1} [ \begin{matrix}q^{−r}, v \\ q^{1−r}/v \end{matrix}; q, q/v^{2}a^{2}]

= ∑_{i=0}^{r} \frac{(u; q)_{r−i}}{(q; q)_{r−i}} H_{r−i} x^{i} \frac{(1/x, uq^{r−i}, v^{2}q^{2r−2i+1}; q)_{i}}{(q, vq^{r−i+1}, xv^{2}q^{2r−2i+1}; q)_{i}} \frac{1 − v^{2}q^{2r}}{1 − v^{2}q^{2r−i}} .

Proof [14]. First observe that the right-hand side is

∑_{i=0}^{r} \frac{(u; q)_{i}}{(q; q)_{i}} \frac{(1/x, uq^{i}, v^{2}q^{2i+1}; q)_{r−i}}{(q, vq^{i+1}, xv^{2}q^{2i+1}; q)_{r−i}} \frac{1 − v^{2}q^{2r}}{1 − v^{2}q^{r+i}} x^{r−i}H_{i}

= x^{r} \frac{(u, 1/x, v^{2}; q)_{r}}{(q, vq, xv^{2}q; q)_{r}} \frac{1 − v^{2}q^{2r}}{1 − v^{2}} ∑_{i=0}^{r} \frac{(q^{−r}, vq, v^{2}q^{r}; q)_{i}}{(q, xq^{1−r}, xv^{2}q^{1+r}; q)_{i}} \frac{(xv^{2}q; q)_{2i}}{(v^{2}q; q)_{2i}} (−q/v)^{i}

× \frac{(vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −xv, xv^{2}; q)_{i}}{(xv^{2}; q)_{2i}} _{4}ϕ_{3} [ \begin{matrix}q^{−i}, xv^{2}q^{i}, −av, −1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −xv \end{matrix}; q, q]

= x^{r} \frac{(u, 1/x, v^{2}, −vq; q)_{r}}{(q, v, −v, xv^{2}q; q)_{r}} ∑_{i=0}^{r} \frac{1 − xv^{2}q^{2i}}{1 − xv^{2}} \frac{(q^{−r}, v^{2}q^{r}, xv^{2}, −xv; q)_{i}}{(q, xq^{1−r}, xv^{2}q^{1+r}, −qv; q)_{i}} (−q/v)^{i}

×_{4}ϕ_{3} [ \begin{matrix}q^{−i}, xv^{2}q^{i}, −av, −1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −xv \end{matrix}; q, q] .

Expanding the _{4}ϕ_{3} series over the index j ≤ i, putting i = j + k, and summing over k, we obtain

x^{r} \frac{(u, 1/x, v^{2}, −vq; q)_{r}}{(q, v, −v, xv^{2}q; q)_{r}} ∑_{j=0}^{r} \frac{(xv^{2}q; q)_{2j}}{(v^{2}q; q)_{2j}} \frac{(vq, −av, −1/a, v^{2}q^{r}, q^{−r}; q)_{j}}{(q, xq^{1−r}, xv^{2}q^{1+r}; q)_{j}} (q/v)^{j} q^{−\binom{j}{2}}

×_{6}ϕ_{5} [ \begin{matrix}xv^{2}q^{2j}, q(xv^{2}q^{2j})^{\frac{1}{2}} , −q(xv^{2}q^{2j})^{\frac{1}{2}} , −xvq^{j}, v^{2}q^{r+j}, q^{j−r} \\ (xv^{2}q^{2j})^{\frac{1}{2}} , −(xv^{2}q^{2j})^{\frac{1}{2}} , −vq^{j+1}, xq^{j−r+1}, xv^{2}q^{j+r+1} \end{matrix} ; q, −q^{1−j}/v] .

By [3, (II.21)] this terminating very-well-poised _{6}ϕ_{5} sum equals

\frac{(xv^{2}q^{2j+1}, −q^{1−r}/v; q)_{r−j}}{(−vq^{j+1}, xq^{j−r+1}; q)_{r−j}} = (\frac{−1}{xv})^{r}v^{j}q^{\binom{j}{2}} \frac{(xv^{2}q, −v; q)_{r}}{(1/x, −vq; q)_{r}} \frac{(−vq, xv^{2}q^{1+r}, xq^{1−r}; q)_{j}}{(−v; q)_{j}(xv^{2}q; q)_{2j}} .

Thus, on simplification, the right-hand side may be written

\frac{(u, v^{2}; q)_{r}}{(q, v; q)_{r}} (−v)^{−r} _{4}ϕ_{3} [ \begin{matrix}q^{−r}, v^{2}q^{r}, −av, −1/a \\ vq^{\frac{1}{2}} , −vq^{\frac{1}{2}} , −v \end{matrix}; q, q] .

Applying Lemma 7 with −a substituted to a, we obtain the left-hand side. □

Acknowledgements

It is a pleasure to thank Alain Lascoux, Mizan Rahman and Michael Schlosser for their interest and generous help.

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Institut Gaspard Monge, Université de Marne-la-Vallée, 77454 Marne-la-Vallée Cedex, France

E-mail address: lassalle@univ-mlv.fr
URL: http://igm.univ-mlv.fr/^{∼}lassalle